Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{x^2 + x - 56}{9x - 9} \div \dfrac{x + 8}{9x - 9} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{x^2 + x - 56}{9x - 9} \times \dfrac{9x - 9}{x + 8} $ First factor the quadratic. $p = \dfrac{(x + 8)(x - 7)}{9x - 9} \times \dfrac{9x - 9}{x + 8} $ Then factor out any other terms. $p = \dfrac{(x + 8)(x - 7)}{9(x - 1)} \times \dfrac{9(x - 1)}{x + 8} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ (x + 8)(x - 7) \times 9(x - 1) } { 9(x - 1) \times (x + 8) } $ $p = \dfrac{ 9(x + 8)(x - 7)(x - 1)}{ 9(x - 1)(x + 8)} $ Notice that $(x - 1)$ and $(x + 8)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ 9\cancel{(x + 8)}(x - 7)(x - 1)}{ 9(x - 1)\cancel{(x + 8)}} $ We are dividing by $x + 8$ , so $x + 8 \neq 0$ Therefore, $x \neq -8$ $p = \dfrac{ 9\cancel{(x + 8)}(x - 7)\cancel{(x - 1)}}{ 9\cancel{(x - 1)}\cancel{(x + 8)}} $ We are dividing by $x - 1$ , so $x - 1 \neq 0$ Therefore, $x \neq 1$ $p = \dfrac{9(x - 7)}{9} $ $p = x - 7 ; \space x \neq -8 ; \space x \neq 1 $